Train timetable during the end-of-service period is crucial for passenger accessibility and operation cost in urban rail transit networks. Existing studies have investigated the last train timetabling problem for improving passenger accessibility. This study investigates a train timetabling problem for the end-of-service period, which concentrates on the coordination of the service ending time on different lines and the last several train timetables. A mixed-integer linear programming model based on a space–time network is proposed to determine the number of train services provided in the end-of-service period while coordinating the timetables of both last and non-last trains, of which the objective function minimizes the number of inaccessible passengers and operation costs. To address the computational challenges, a Benders decomposition algorithm is developed and enhanced with dedicated acceleration strategies. A dual solution algorithm is proposed to efficiently generate the optimal dual solution of the subproblems. A reformulation and update strategy is proposed for the Benders cuts, and a relax-and-fix heuristic is developed to improve solving efficiency of the master problem. Small-scale numerical experiments demonstrate the optimality and efficiency of the proposed Benders decomposition algorithm. Large-scale experiments in the Wuhan network show that the proposed model and algorithm can improve passenger accessibility by 6.8% without additional operation cost, and by 38.7% with a 28.4% increment in operation cost.

中文翻译：

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服务终止期间的城市轨道列车时刻表以及乘客可达性和运营成本：一种先进的 benders 分解算法


停运期间的列车时刻表对于城市轨道交通网络的乘客可达性和运营成本至关重要。现有研究调查了最后一班火车时刻表问题，以改善乘客的可达性。本研究调查了服务终止期间的列车时刻表问题，该问题集中在不同线路的服务结束时间和最后几趟列车时刻表的协调上。该文提出一种基于时空网络的混合整数线性规划模型，用于确定服务终止期间提供的列车服务数量，同时协调末班车和非末班车的时刻表，其中目标函数使无法到达的乘客数量和运营成本最小化。为了解决计算挑战，开发了 Benders 分解算法，并使用专用的加速策略进行了增强。提出了一种对偶解算法，以有效地生成子问题的最优对偶解。为 Benders 割平面提出了一种重新制定和更新策略，并开发了一种 relax-and-fix 启发式方法，以提高主问题的求解效率。小规模数值实验证明了所提出的 Benders 分解算法的最优性和效率。在武汉网络中的大规模实验表明，所提出的模型和算法可以在不增加运营成本的情况下将乘客可达性提高 6.8%，在运营成本增加 28.4% 的情况下可以提高 38.7%。